DOCSLIB.ORG

Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions Improved Analysis[1] Daniel Bahrdt ∗ Martin P

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions Improved Analysis[1] Daniel Bahrdt ∗ Martin P Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions Improved Analysis[1] Daniel Bahrdt ∗ Martin P. Seybold∗ July 27, 2017 Abstract 1. Introduction Each non-zero point in Rd identifies a closest point x on d−1 the unit sphere S . We are interested in computing an Many mathematical sciences use trigonometric functions d d−1 "-approximation y 2 Q for x, that is exactly on S in symbolic coordinate transformations to simplify fun- and has low bit size. We revise lower bounds on rational damental equations of physics or mathematical systems. approximations and provide explicit, spherical instances. However, rational numbers are dominating in computer We prove that floating-point numbers can only pro- processing as they allow for simple storage as well as 2 vide trivial solutions to the sphere equation in R and fast exact and inexact arithmetics (e.g. GMP[12], IEEE 3 R . Moreover, we show how to construct a rational point Float, MPFR[11]). Therefore problems on spherical sur- 2 with denominators of at most 10(d − 1)=" for any given faces often require to scale a point vector, as in choosing 1 " 2 0; 8 , improving on a previous result. The method a point uniform at random[20], or to evaluate a trigono- further benefits from algorithms for simultaneous Dio- metric function for a rational angle argument, as in deal- phantine approximation. ing with Geo-referenced data. Our open-source implementation and experiments A classical theoretical barrier is Niven's theorem[21], demonstrate the practicality of our approach in the con- which states that the sole rational values of sine for ratio- text of massive data sets Geo-referenced by latitude and nal multiplies of π are 0; ±1=2 and ±1. The well known longitude values. Chebyshev polynomials have roots at these values, hence Keywords Diophantine approximation, Rational give rise to representations for these algebraic numbers. points, Unit sphere, Perturbation, Stable geometric However, arithmetics in a full algebraic number field constructions might well be too demanding for many applications. For products of sine and cosine, working with Euler's formula on the complex unit circle and Chebyshev polynomials would suffice though. This manifests in problems of exact geometrical com- putations, since standard methodology relies on Carte- sian input[18]. Spheres and ellipsoids are common ge- ometric objects and rational solutions to their defining arXiv:1707.08549v1 [cs.CG] 26 Jul 2017 quadratic polynomials are closely related to Diophantine equations of degree 2. The famous Pythagorean Triples are known to identify the rational points on the circle S1. Moreover, the unit sphere has a dense set of ratio- nal points and so do ellipsoids with rational half-axes through scaling. Spherical coordinates are convenient to reference such Cartesians with angle coordinates and geo-referenced data denotes points with rational angles. Figure 1: Spherical Delaunay triangulation (gray) con- Standard approximations of Cartesians do not necessar- strained to contain all line segments (black) of ily fulfill these equations, therefore subsequent algorith- streets in Ecuador and the intersection points mic results can suffer greatly. of constraints (red). This paper focuses on finding rational points exactly d−1 d P 2 on the unit sphere S = x 2 R : i xi = 1 with ∗Formale Methoden der Informatik, University of Stuttgart, Ger- bounded distance to the point x=kxk2 { its closest point many, fbahrdt, [email protected] on Sd−1. In this work, x 2 Rd can be given by any finite 1 means that allow to compute a rational approximation to 1.2. Contribution it with arbitrary target precision. Using rational Carte- The strong lower bound on rational approximations to sian approximations for spherical coordinates, as derived other rational values does not hold for Geo-referenced from MPFR, is just one example of such a black-box data considering Niven's theorem. We derive explicit model. Moreover, we are interested in calculating ratio- constants to Liouville's lower bound, for a concrete nal points on d with small denominators. S Geo-referenced point, that is within a factor 2 of the strong lower bound. Moreover, we prove that floating- point numbers cannot represent Cartesian coordinates of 1.1. Related Work points that are exactly on S1 or S2. We describe how the use of rotation symmetry and Studies on spherical Delaunay triangulations (SDT), us- approximations with fixed-point numbers suffice to im- ing great-circle segments on the sphere S2, provide com- prove on the main theorem of [28]. We derive rational mon ways to avoid and deal with the point-on-sphere points exactly on Sd−1 with denominators of at most problem in computational geometry. 2 1 10(d−1)=" for any " 2 0; 8 . Moreover, our method al- The fragile approaches [23, 14, 25] ignore that the in- lows for even smaller denominators based on algorithms put may not be on S2 and succeed if the results of all for simultaneous Diophantine approximations, though a predicate evaluations happen to be correct. Input point potentially weaker form of approximation would suffice. arrangements with close proximity or unfortunate loca- The controlled perturbations provided by our method tions bring these algorithms to crash, loop or produce allow exact geometric algorithms on Sd to rely on ra- erroneous output. The quasi-robust approaches [5,4] tional rather than algebraic numbers { E.g. enabling weaken the objective and calculate a Delaunay tessel- convex hull algorithms to efficiently obtain spherical De- lation in d-Simplexes. Lifting to a d + 1 convex hull launay triangulations on Sd and not just Delaunay tes- problem is achieved by augmenting a rational coordinate sellations. Moreover, the approach allows for inexact but from a quadratic form { The augmented point exactly "-stable geometric constructions { E.g. intersections of meets the (elliptic) paraboloid equation. However, the Great Circle segments. output only identifies a SDT if all input points are al- We demonstrate the quality and effectiveness of the ready on the sphere, otherwise the objectives are distinct. method on several, including one whole-world sized, Equally unclear is how to address spherical predicates point sets. We provide open-source implementations for and spherical constructions. The robust approaches [27] the method and its application in the case of spherical use the circle preserving stereographic projection from Delaunay triangulations with intersections of constraints. S2 to the plane. The perturbation to input, for which the output is correct, can be very large as the projec- tion does not preserve distances. Furthermore, achieving 2. Definitions and Tools additional predicates and constructions remains unclear. The 2nd Chebyshev polynomials U of degree n are in The stable approaches provide geometric predicates and n Z[X], given their recursive definition: constructions for points on S2 by explicitly storing an al- gebraic number, originating from scaling an ordinary ra- U0(x) = 1 U1(x) = 2x tional approximation to unit length[10]. Algebraic num- Un+1(x) = 2xUn(x) − Un−1(x) : ber arithmetics can be avoided for S2, but exact evalua- tion relies on specifically tailored predicates [8], leaving It is well known [26], that the n roots of Un are exactly the implementation of new constructions and predicates the values f cos (πk= (n + 1)) : k = 1; : : : ; n g. Hence open. the polynomials Un give rise to algebraic representations Kleinbock and Merrill provide methods to quantify the for cosine values of rational multiplies of π. This is par- density of rational points on Sd [16], that extend to other ticularly useful in conjunction with classic results on Dio- manifolds as well. Recently, Schmutz[28] provided an phantine approximations, that are known since 1844[19]: divide-&-conquer approach on the sphere equation, us- ing Diophantine approximation by continued fractions, Theorem 1 (Liouville's Lower Bound). For any alge- to derive points in Qd \ Sd−1 for a point on the unit braic α 2 R of degree n ≥ 2, there is a positive constant sphere Sd−1. The main theorem bounds the denomina- c(α) > 0 such that tors in "-approximations, under the k k norm, with p 1 p c(α) ( 32dlog de=")2dlog2 de. Based on this, rational approxi- α − ≥ 2 q qn mations in the orthogonal group O(n; R) and in the uni- tary matrix group U(n; ) are found. This is of particu- C for any p 2 Z and q 2 N. lar interest for sweep-line algorithms: [7] studies finding a rotation matrix with small rationals for a given rational Apart from this lower bound on rational approxima- rotation angle of an 2D arrangement. tions, there is another important folklore result on the 2 a p existence of simultaneous Diophantine approximations. Observation 1. For rational numbers b 6= q , we have Such approximations have surprisingly small errors, de- spite their rather small common denominator. a p aq − bp 1 − = ≥ b q bq bq Theorem 2 (Dirichlet's Upper Bound). Let N 2 N and α 2 d with 0 ≤ α ≤ 1. There are integers p 2 d; q 2 R i Z 2 with 1 ≤ q ≤ N and If q < b, we have a lower bound of 1=q for ratio- Z a nal approximations to b with denominators up to q. 1 3 α − pi ≤ p : Pythagorean triples (x; y; z) 2 N provide such ratio- i q d 1 2 2 q N nal points on S , since (x=z) + (y=z) = 1. We have a lower bound of 1=z2 for approximations with denomina- (See AppendixB for a proof.) For d = 1, the continued tors q < z. See Section 3.4 for rational points on Sd with fraction (equivalently the Euclidean) algorithm is famous the same denominator property.
Load more

Recommended publications
  • Spheres in Infinite-Dimensional Normed Spaces Are Lipschitz Contractible
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 88. Number 3, July 1983 SPHERES IN INFINITE-DIMENSIONAL NORMED SPACES ARE LIPSCHITZ CONTRACTIBLE Y. BENYAMINI1 AND Y. STERNFELD Abstract. Let X be an infinite-dimensional normed space. We prove the following: (i) The unit sphere {x G X: || x II = 1} is Lipschitz contractible. (ii) There is a Lipschitz retraction from the unit ball of JConto the unit sphere. (iii) There is a Lipschitz map T of the unit ball into itself without an approximate fixed point, i.e. inffjjc - Tx\\: \\x\\ « 1} > 0. Introduction. Let A be a normed space, and let Bx — {jc G X: \\x\\ < 1} and Sx = {jc G X: || jc || = 1} be its unit ball and unit sphere, respectively. Brouwer's fixed point theorem states that when X is finite dimensional, every continuous self-map of Bx admits a fixed point. Two equivalent formulations of this theorem are the following. 1. There is no continuous retraction from Bx onto Sx. 2. Sx is not contractible, i.e., the identity map on Sx is not homotopic to a constant map. It is well known that none of these three theorems hold in infinite-dimensional spaces (see e.g. [1]). The natural generalization to infinite-dimensional spaces, however, would seem to require the maps to be uniformly-continuous and not merely continuous. Indeed in the finite-dimensional case this condition is automatically satisfied. In this article we show that the above three theorems fail, in the infinite-dimen- sional case, even under the strongest uniform-continuity condition, namely, for maps satisfying a Lipschitz condition.
    [Show full text]
  • Examples of Manifolds
    Examples of Manifolds Example 1 (Open Subset of IRn) Any open subset, O, of IRn is a manifold of dimension n. One possible atlas is A = (O, ϕid) , where ϕid is the identity map. That is, ϕid(x) = x. n Of course one possible choice of O is IR itself. Example 2 (The Circle) The circle S1 = (x,y) ∈ IR2 x2 + y2 = 1 is a manifold of dimension one. One possible atlas is A = {(U , ϕ ), (U , ϕ )} where 1 1 1 2 1 y U1 = S \{(−1, 0)} ϕ1(x,y) = arctan x with − π < ϕ1(x,y) <π ϕ1 1 y U2 = S \{(1, 0)} ϕ2(x,y) = arctan x with 0 < ϕ2(x,y) < 2π U1 n n n+1 2 2 Example 3 (S ) The n–sphere S = x =(x1, ··· ,xn+1) ∈ IR x1 +···+xn+1 =1 n A U , ϕ , V ,ψ i n is a manifold of dimension . One possible atlas is 1 = ( i i) ( i i) 1 ≤ ≤ +1 where, for each 1 ≤ i ≤ n + 1, n Ui = (x1, ··· ,xn+1) ∈ S xi > 0 ϕi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n Vi = (x1, ··· ,xn+1) ∈ S xi < 0 ψi(x1, ··· ,xn+1)=(x1, ··· ,xi−1,xi+1, ··· ,xn+1) n So both ϕi and ψi project onto IR , viewed as the hyperplane xi = 0. Another possible atlas is n n A2 = S \{(0, ··· , 0, 1)}, ϕ , S \{(0, ··· , 0, −1)},ψ where 2x1 2xn ϕ(x , ··· ,xn ) = , ··· , 1 +1 1−xn+1 1−xn+1 2x1 2xn ψ(x , ··· ,xn ) = , ··· , 1 +1 1+xn+1 1+xn+1 are the stereographic projections from the north and south poles, respectively.
    [Show full text]
  • Convolution on the N-Sphere with Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE
    IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 1157 Convolution on the n-Sphere With Application to PDF Modeling Ivan Dokmanic´, Student Member, IEEE, and Davor Petrinovic´, Member, IEEE Abstract—In this paper, we derive an explicit form of the convo- emphasis on wavelet transform in [8]–[12]. Computation of the lution theorem for functions on an -sphere. Our motivation comes Fourier transform and convolution on groups is studied within from the design of a probability density estimator for -dimen- the theory of noncommutative harmonic analysis. Examples sional random vectors. We propose a probability density function (pdf) estimation method that uses the derived convolution result of applications of noncommutative harmonic analysis in engi- on . Random samples are mapped onto the -sphere and esti- neering are analysis of the motion of a rigid body, workspace mation is performed in the new domain by convolving the samples generation in robotics, template matching in image processing, with the smoothing kernel density. The convolution is carried out tomography, etc. A comprehensive list with accompanying in the spectral domain. Samples are mapped between the -sphere theory and explanations is given in [13]. and the -dimensional Euclidean space by the generalized stereo- graphic projection. We apply the proposed model to several syn- Statistics of random vectors whose realizations are observed thetic and real-world data sets and discuss the results. along manifolds embedded in Euclidean spaces are commonly termed directional statistics. An excellent review may be found Index Terms—Convolution, density estimation, hypersphere, hy- perspherical harmonics, -sphere, rotations, spherical harmonics. in [14]. It is of interest to develop tools for the directional sta- tistics in analogy with the ordinary Euclidean.
    [Show full text]
  • Minkowski Products of Unit Quaternion Sets 1 Introduction
    Minkowski products of unit quaternion sets 1 Introduction The Minkowski sum A⊕B of two point sets A; B 2 Rn is the set of all points generated [16] by the vector sums of points chosen independently from those sets, i.e., A ⊕ B := f a + b : a 2 A and b 2 B g : (1) The Minkowski sum has applications in computer graphics, geometric design, image processing, and related fields [9, 11, 12, 13, 14, 15, 20]. The validity of the definition (1) in Rn for all n ≥ 1 stems from the straightforward extension of the vector sum a + b to higher{dimensional Euclidean spaces. However, to define a Minkowski product set A ⊗ B := f a b : a 2 A and b 2 B g ; (2) it is necessary to specify products of points in Rn. In the case n = 1, this is simply the real{number product | the resulting algebra of point sets in R1 is called interval arithmetic [17, 18] and is used to monitor the propagation of uncertainty through computations in which the initial operands (and possibly also the arithmetic operations) are not precisely determined. A natural realization of the Minkowski product (2) in R2 may be achieved [7] by interpreting the points a and b as complex numbers, with a b being the usual complex{number product. Algorithms to compute Minkowski products of complex{number sets have been formulated [6], and extended to determine Minkowski roots and powers [3, 8] of complex sets; to evaluate polynomials specified by complex{set coefficients and arguments [4]; and to solve simple equations expressed in terms of complex{set coefficients and unknowns [5].
    [Show full text]
  • GEOMETRY Contents 1. Euclidean Geometry 2 1.1. Metric Spaces 2 1.2
    GEOMETRY JOUNI PARKKONEN Contents 1. Euclidean geometry 2 1.1. Metric spaces 2 1.2. Euclidean space 2 1.3. Isometries 4 2. The sphere 7 2.1. More on cosine and sine laws 10 2.2. Isometries 11 2.3. Classification of isometries 12 3. Map projections 14 3.1. The latitude-longitude map 14 3.2. Stereographic projection 14 3.3. Inversion 14 3.4. Mercator’s projection 16 3.5. Some Riemannian geometry. 18 3.6. Cylindrical projection 18 4. Triangles in the sphere 19 5. Minkowski space 21 5.1. Bilinear forms and Minkowski space 21 5.2. The orthogonal group 22 6. Hyperbolic space 24 6.1. Isometries 25 7. Models of hyperbolic space 30 7.1. Klein’s model 30 7.2. Poincaré’s ball model 30 7.3. The upper halfspace model 31 8. Some geometry and techniques 32 8.1. Triangles 32 8.2. Geodesic lines and isometries 33 8.3. Balls 35 9. Riemannian metrics, area and volume 36 Last update: December 12, 2014. 1 1. Euclidean geometry 1.1. Metric spaces. A function d: X × X ! [0; +1[ is a metric in the nonempty set X if it satisfies the following properties (1) d(x; x) = 0 for all x 2 X and d(x; y) > 0 if x 6= y, (2) d(x; y) = d(y; x) for all x; y 2 X, and (3) d(x; y) ≤ d(x; z) + d(z; y) for all x; y; z 2 X (the triangle inequality). The pair (X; d) is a metric space.
    [Show full text]
  • Lp Unit Spheres and the Α-Geometries: Questions and Perspectives
    entropy Article Lp Unit Spheres and the a-Geometries: Questions and Perspectives Paolo Gibilisco Department of Economics and Finance, University of Rome “Tor Vergata”, Via Columbia 2, 00133 Rome, Italy; [email protected] Received: 12 November 2020; Accepted: 10 December 2020; Published: 14 December 2020 Abstract: In Information Geometry, the unit sphere of Lp spaces plays an important role. In this paper, the aim is list a number of open problems, in classical and quantum IG, which are related to Lp geometry. Keywords: Lp spheres; a-geometries; a-Proudman–Johnson equations Gentlemen: there’s lots of room left in Lp spaces. 1. Introduction Chentsov theorem is the fundamental theorem in Information Geometry. After Rao’s remark on the geometric nature of the Fisher Information (in what follows shortly FI), it is Chentsov who showed that on the simplex of the probability vectors, up to scalars, FI is the unique Riemannian geometry, which “contract under noise” (to have an idea of recent developments about this see [1]). So FI appears as the “natural” Riemannian geometry over the manifolds of density vectors, namely over 1 n Pn := fr 2 R j ∑ ri = 1, ri > 0g i Since FI is the pull-back of the map p r ! 2 r it is natural to study the geometries induced on the simplex of probability vectors by the embeddings 8 1 <p · r p p 2 [1, +¥) Ap(r) = :log(r) p = +¥ Setting 2 p = a 2 [−1, 1] 1 − a we call the corresponding geometries on the simplex of probability vectors a-geometries (first studied by Chentsov himself).
    [Show full text]
  • 17 Measure Concentration for the Sphere
    17 Measure Concentration for the Sphere In today’s lecture, we will prove the measure concentration theorem for the sphere. Recall that this was one of the vital steps in the analysis of the Arora-Rao-Vazirani approximation algorithm for sparsest cut. Most of the material in today’s lecture is adapted from Matousek’s book [Mat02, chapter 14] and Keith Ball’s lecture notes on convex geometry [Bal97]. n n−1 Notation: We will use the notation Bn to denote the ball of unit radius in R and S n to denote the sphere of unit radius in R . Let µ denote the normalized measure on the unit sphere (i.e., for any measurable set S ⊆ Sn−1, µ(A) denotes the ratio of the surface area of µ to the entire surface area of the sphere Sn−1). Recall that the n-dimensional volume of a ball n n n of radius r in R is given by the formula Vol(Bn) · r = vn · r where πn/2 vn = n Γ 2 + 1 Z ∞ where Γ(x) = tx−1e−tdt 0 n−1 The surface area of the unit sphere S is nvn. Theorem 17.1 (Measure Concentration for the Sphere Sn−1) Let A ⊆ Sn−1 be a mea- n−1 surable subset of the unit sphere S such that µ(A) = 1/2. Let Aδ denote the δ-neighborhood n−1 n−1 of A in S . i.e., Aδ = {x ∈ S |∃z ∈ A, ||x − z||2 ≤ δ}. Then, −nδ2/2 µ(Aδ) ≥ 1 − 2e . Thus, the above theorem states that if A is any set of measure 0.5, taking a step of even √ O (1/ n) around A covers almost 99% of the entire sphere.
    [Show full text]
  • On Expansive Mappings
    mathematics Article On Expansive Mappings Marat V. Markin and Edward S. Sichel * Department of Mathematics, California State University, Fresno 5245 N. Backer Avenue, M/S PB 108, Fresno, CA 93740-8001, USA; [email protected] * Correspondence: [email protected] Received: 23 August 2019; Accepted: 17 October 2019; Published: 23 October 2019 Abstract: When finding an original proof to a known result describing expansive mappings on compact metric spaces as surjective isometries, we reveal that relaxing the condition of compactness to total boundedness preserves the isometry property and nearly that of surjectivity. While a counterexample is found showing that the converse to the above descriptions do not hold, we are able to characterize boundedness in terms of specific expansions we call anticontractions. Keywords: Metric space; expansion; compactness; total boundedness MSC: Primary 54E40; 54E45; Secondary 46T99 O God, I could be bounded in a nutshell, and count myself a king of infinite space - were it not that I have bad dreams. William Shakespeare (Hamlet, Act 2, Scene 2) 1. Introduction We take a close look at the nature of expansive mappings on certain metric spaces (compact, totally bounded, and bounded), provide a finer classification for such mappings, and use them to characterize boundedness. When finding an original proof to a known result describing all expansive mappings on compact metric spaces as surjective isometries [1] (Problem X.5.13∗), we reveal that relaxing the condition of compactness to total boundedness still preserves the isometry property and nearly that of surjectivity. We provide a counterexample of a not totally bounded metric space, on which the only expansion is the identity mapping, demonstrating that the converse to the above descriptions do not hold.
    [Show full text]
  • Visualization of Quaternions with Clifford Parallelism
    Proceedings of the 20th Asian Technology Conference in Mathematics (Leshan, China, 2015) Visualization of quaternions with Clifford parallelism Yoichi Maeda [email protected] Department of Mathematics Tokai University Japan Abstract: In this paper, we try to visualize multiplication of unit quaternions with dynamic geometry software Cabri 3D. The three dimensional sphere is identified with the set of unit quaternions. Multiplication of unit quaternions is deeply related with Clifford parallelism, a special isometry in . 1. Introduction The quaternions are a number system that extends the complex numbers. They were first described by William R. Hamilton in 1843 ([1] p.186, [3]). The identities 1, where , and are basis elements of , determine all the possible products of , and : ,,,,,. For example, if 1 and 23, then, 12 3 2 3 2 3. We are very used to the complex numbers and we have a simple geometric image of the multiplication of complex numbers with the complex plane. How about the multiplication of quaternions? This is the motivation of this research. Fortunately, a unit quaternion with norm one (‖‖ 1) is regarded as a point in . In addition, we have the stereographic projection from to . Therefore, we can visualize three quaternions ,, and as three points in . What is the geometric relation among them? Clifford parallelism plays an important role for the understanding of the multiplication of quaternions. In section 2, we review the stereographic projection. With this projection, we try to construct Clifford parallel in Section 3. Finally, we deal with the multiplication of unit quaternions in Section 4. 2. Stereographic projection of the three-sphere The stereographic projection of the two dimensional sphere ([1, page 260], [2, page 74]) is a very important map in mathematics.
    [Show full text]
  • Metric Spaces and Continuity This Publication Forms Part of an Open University Module
    M303 Further pure mathematics Metric spaces and continuity This publication forms part of an Open University module. Details of this and other Open University modules can be obtained from the Student Registration and Enquiry Service, The Open University, PO Box 197, Milton Keynes MK7 6BJ, United Kingdom (tel. +44 (0)845 300 6090; email [email protected]). Alternatively, you may visit the Open University website at www.open.ac.uk where you can learn more about the wide range of modules and packs offered at all levels by The Open University. Note to reader Mathematical/statistical content at the Open University is usually provided to students in printed books, with PDFs of the same online. This format ensures that mathematical notation is presented accurately and clearly. The PDF of this extract thus shows the content exactly as it would be seen by an Open University student. Please note that the PDF may contain references to other parts of the module and/or to software or audio-visual components of the module. Regrettably mathematical and statistical content in PDF files is unlikely to be accessible using a screenreader, and some OpenLearn units may have PDF files that are not searchable. You may need additional help to read these documents. The Open University, Walton Hall, Milton Keynes, MK7 6AA. First published 2014. Second edition 2016. Copyright c 2014, 2016 The Open University All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, transmitted or utilised in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without written permission from the publisher or a licence from the Copyright Licensing Agency Ltd.
    [Show full text]
  • Chapters 4 (Pdf)
    This is page 142 Printer: Opaque this This is page 143 Printer: Opaque this CHAPTER 4 FORMS ON MANIFOLDS 4.1 Manifolds Our agenda in this chapter is to extend to manifolds the results of Chapters 2 and 3 and to formulate and prove manifold versions of two of the fundamental theorems of integral calculus: Stokes’ theorem and the divergence theorem. In this section we’ll define what we mean by the term “manifold”, however, before we do so, a word of encouragement. Having had a course in multivariable calculus, you are already familiar with manifolds, at least in their one and two dimensional emanations, as curves and surfaces in R3, i.e., a manifold is basically just an n-dimensional surface in some high dimensional Euclidean space. To make this definition precise let X be a subset of RN , Y a subset of Rn and f : X Y a continuous map. We recall → Definition 4.1.1. f is a ∞ map if for every p X, there exists a C ∈ neighborhood, U , of p in RN and a ∞ map, g : U Rn, which p C p p → coincides with f on U X. p ∩ We also recall: Theorem 4.1.2. If f : X Y is a ∞ map, there exists a neigh- → C borhood, U, of X in RN and a ∞ map, g : U Rn such that g C → coincides with f on X. (A proof of this can be found in Appendix A.) We will say that f is a diffeomorphism if it is one–one and onto and f and f −1 are both ∞ maps.
    [Show full text]
  • Ali-Manifolds.Pdf
    Basic notions of Differential Geometry Md. Showkat Ali 7th November 2002 Manifold Manifold is a multi dimensional space. Riemann proposed the concept of ”multiply extended quantities” which was refined for the next fifty years and is given today by the notion of a manifold. Definition. A Hausdorff topological space M is called an n-dimensional (topological) manifold (denoted by M n) if every point in M has an open neighbourhood which is homeomorphic to an open set in Rn. Example 1. A surface R2 is a two-dimensional manifold, since φ : R2 R2 is homeomorphic. −→ Charts, Ck-class and Atlas Definition. A pair (U, φ) of a topological manifold M is an open subset U of M called the domain of the chart together with a homeomorphism φ : U V of U onto an open set V in Rn. Roughly−→ speaking, a chart of M is an open subset U in M with each point in U labelled by n-numbers. Definition. A map f : U Rm is said to be of class Ck if every func- tion f i(x1, ..., xn), i = 1, ...,−→ m is differentiable upto k-th order. The map m f : U R is called smooth if it is of C∞-class. −→ Definition. An atlas of class Ck on a topological manifold M is a set (Uα, φα), α I of each charts, such that { ∈ } (i) the domain Uα covers M, i.e. M = αUα, ∪ (ii) the homeomorphism φα satisfy the following compativility condition : 1 the map 1 φα φ− : φβ(Uα Uβ) φα(Uα Uβ) ◦ β ∩ −→ ∩ must be of class Ck.
    [Show full text]